If $a+b$ and $ab$ are algebraic then $a,b$ are algebraic

Asked Apr 18 '15 at 21:46

Active Apr 09 '16 at 12:01

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I need to show that if the sum and the product of two complex numbers is algebraic, then each of then is algebraic.

We have that the extensions $Q(a+b)/Q$ and $Q(ab)/Q$ are finite, so the extension $Q(a+b,ab)/Q$ is finite.

Since $$[Q(a,b):Q]=[Q(a,b):Q(a+b,ab)] [Q(a+b,ab):Q],$$ it suffices to show that the extension $Q(a,b)/Q(a+b,ab)$ is finite, but I can't come up with an appropriate basis. Any ideas?

edited Apr 09 '16 at 12:01

choco_addicted

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asked Apr 18 '15 at 21:46

user232536

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What are the roots of $X^2 - (a+b)X+ab = 0$? – Tryss Apr 18 '15 at 21:51
The roots are $-(a+b) \pm \sqrt{a^2+b^2}$. These are in Q(a+b,ab), but I'm afraid I don't see what your point is – user232536 Apr 18 '15 at 21:59
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No, the roots of this polynomial are exacly $a$ and $b$ – Tryss Apr 18 '15 at 22:00
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Well, now that you changed it they are :P but thanks! – user232536 Apr 18 '15 at 22:08

If $a+b$ and $ab$ are algebraic then $a,b$ are algebraic

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